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Let’s Bring Back That Gee-om-met-tree!

By Katharine Merow

Richard Guy’s talk at MAA MathFest in August began with a cautionary note: “WARNING!!” read Guy’s second slide, “This presentation contains scenes with Complicated Configurations and a Notable Need for a Nice Notation.”

Weeks shy of his 99th birthday, Guy (University of Calgary, emeritus) delivered his lecture, “A Triangle Has Eight Vertices but Only One Centre,” as part of a Special Invited Session called “The Geometry of Triangles.”

Guy walked his audience through the geometric manipulations by which a triangle gives rise to symmetric sets of points, lines, and circles. There are the titular vertices, yes, but also a plethora of other named entities: Gergonne (http://bit.ly/1K8kKOg) and de Longchamps (http://bit.ly/1JsTN9f) points, Euler (http://bit.ly/1Ll0jfK) and Simson-Wallace (http://bit.ly/1LmJwMc) lines, and what Guy termed the “Steiner Star of David.”

“There are an infinite number of things you can say about a triangle,” Guy remarked.

The session concluded with those assembled belting out—to the tune of “My Bonnie Lies over the Ocean”—lyrics Guy had composed ahead of the summer conference season. (Guy’s talk at the Mathematics of Very Entertaining Subjects (MOVES) meeting in New York City in August (http://bit.ly/1WGP87v) also ended in song.)

Quadration: The intersection of a triangle’s three altitudes is its orthocenter. Quadration grants the orthocenter equal status with the triangle’s three vertices. (Convince yourself that each of the four points is the orthocenter of the triangle formed by the other three.) Under quadration, then, a triangle is an orthocentric quadrangle (http://bit.ly/1Jkw5Wy).

Twinning: A circumcircle is the unique circle that passes through each of a triangle’s three vertices, and the center of the circumcircle is called the circumcenter. If you draw the perpendicular bisectors of each of the six edges of an orthocentric quadrangle, Guy says, you produce a quadrangle of circumcenters that is congruent to the original quadrangle. Guy calls the production of this second quadrangle twinning.

Extraversion: Extraversion is John Conway’s word for the study of what happens to theorems in triangle geometry as you smoothly move two vertices A and C of a triangle ABC through each other. (See a nice animation at http://bit.ly/1gYNA82.) This movement causes the incircle (or inscribed circle) of the original ABC to change places with the b-excircle (see http://bit.ly/1HY74zi for a definition of excircle). And for any algebraic result about the incircle or incenter, a corresponding result holds for the b-excircle or excenter as long as you change the sign of b. (The incenter and excenter are the centers of the incircle and excircle, respectively.)

“There’s a pun, of course,” Conway said of extraversion in his MathFest talk (which followed Guy’s), “since I invented the term.” Extraversion involves “extraverting” a triangle or turning it inside out, Conway explained, but it also produces “extra versions” of various entities.